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Understanding the Guiding of Kinesin/Microtubule-Based Microtransporters in Microfabricated Tracks Yuki Ishigure† and Takahiro Nitta*,†,‡ †

Department of Mathematical Design and Engineering and ‡Applied Physics Course, Gifu University, Gifu 501-1193, Japan S Supporting Information *

ABSTRACT: Microtransporters using cargo-laden microtubules propelled by kinesin motors are attractive for numerous applications in nanotechnology. To improve the efficiency of transport, the movement of microtubules must be guided by microfabricated tracks. However, the mechanisms of the guiding methods used are not fully understood. Here, using computer simulation, we systematically studied the guiding of such microtransporters by three different types of guiding methods: a chemical boundary, a physical barrier, and their combination. The simulation reproduced the probabilities of guiding previously observed experimentally for the three methods. Moreover, the simulation provided further insight into the mechanisms of guiding, which overturn previous assumptions and models.

1. INTRODUCTION Linear motor proteins are ubiquitous in biology and are involved in intracellular transport (e.g., kinesin-1, hereafter kinesin) and muscle contractions (myosin-II). For instance, kinesin “walks” along microtubules (MTs) with a hand-overhand mechanism, taking 8 nm steps by hydrolyzing adenosine triphosphate (ATP). Because of their ability to efficiently convert chemical energy to mechanical energy as well as their small size, motor proteins are desirable for various applications in nanotechnology.1−11 One such application is a so-called molecular shuttle (MS) driven by motor proteins.12 By immobilizing the motors on surfaces, cargo-laden MTs or actin filaments are shuttled around, acting as microtransporters. These MSs have several desirable qualities: (1) they can work in environments relevant to biomedical and biotechnological applications;13 (2) they can be fueled by ATP, which is abundant in biological media or can be easily added to the solution; and (3) they are propelled with large forces (up to several hundred piconewtons), which enables the transport of large entities such as bacterial cells14 and microfabricated objects.15,16 Thus, MSs can be utilized for surface imaging,17,18 wiring for DNA-based nanoelectronics,19 biocomputation,20 and molecular communication.21,22 Among these applications, biosensors integrated with MSs have been investigated most extensively, and prototypical devices have been developed.23−26 To improve the efficiency of MS transport (e.g., transportation time), MS movements must be guided. For example, without guiding tracks, the detection time of biosensors integrated with MSs is comparable to those that rely on surface diffusion. However, with appropriate guiding tracks, the detection time can be shortened by orders of magnitude and does not suffer from mass transport limitations.27 Such guiding © 2014 American Chemical Society

is commonly achieved through the use of microfabricated guiding tracks.28−33 For example, microfabricated walls made by photolithography or electron beam lithography are often used to guide MS movements by physically restricting the regions in which MSs can move. In spite of the importance of MS guiding, the mechanisms through which guiding can be achieved are not fully understood. Although analytical models have provided insight into guiding events, their assumptions have not been well examined.30,34−36 In addition, these theories were developed for ideal conditions, which may not be achieved experimentally. For example, perfect nonfouling surfaces have not yet been created. Hence, in experiments, kinesin motors may adhere to nonfouling surfaces. However, analytical models that deal with more realistic conditions are limited in their applicability. Furthermore, Sundberg et al. reported a discrepancy between the theoretically predicted and experimentally obtained probabilities of guiding by chemically defined guiding tracks,35 which calls into question the validity of the modeling. Computer simulations are useful in understanding MS guiding for several reasons: (1) they can be performed for a range of different applications; (2) they allow for precisely controlled conditions; and (3) they provide analysis with high spatiotemporal resolution. We have previously developed a computer simulation for designing guiding track layouts.37,38 To simulate hundreds of MSs, the simulation produced trajectories only of the leading tips of MSs rather than entire MTs or actin filaments using experimentally determined Received: June 6, 2014 Revised: September 5, 2014 Published: September 19, 2014 12089

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parameters.39 This simulation was shown to be useful for the in silico design and testing of guiding track layouts,26,37,38,40,41 investigating the mechanism of motor-driven self-assembly,42 testing hypotheses of shuttle movement,43,44 rationalizing the formation of unipolar MT arrays,45 and investigating the transmission of information in molecular communication.22 However, owing to the nature of this simulation method, detailed guiding mechanisms could not be investigated. To elucidate the guiding mechanism in detail, cytoskeletal filaments (MTs or actin filaments), motor proteins (kinesin or myosin), and the microfabricated walls must be explicitly modeled. Currently, there are simulations that explicitly model cytoskeletons and their associated motor proteins,46 and some of these simulations have been used to investigate the guiding. For example, Chen et al. used a 2D simulation to study interactions between MTs propelled by dynein motors and between MTs and guiding walls.47 Nedelec and colleagues developed a simulation package48 and used this to design guiding tracks, which were created by patterns of kinesin and hypothetical minus-end-directed motors on planar surfaces with various functions, such as directed transport and polarity sorting.49 In addition, they also studied guiding events performed by chemical edges with their 2D simulation.49 However, no simulation of the 3D movements involved in MS guiding has been performed. Here, using a computer simulation that reproduces 3D MS movements, we systematically studied the guiding of MSs based on kinesin and MTs for the following three types of guiding track boundaries: (1) a chemical edge between kinesin-coated and kinesin-free bottom surfaces; (2) microfabricated walls coated with kinesin; and (3) nonfouling microfabricated walls next to kinesin-coated bottom surfaces. The mechanisms governing guiding and interpretations of the experimental results previously obtained are discussed.

Figure 1. Schematic representation of the simulation method. (a) Overview of the simulation. A microtubule propelled by kinesin motors approaches the boundary of a microfabricated track with an approach angle of θ. In this figure, a perfect nonfouling wall with a height of h and steepness of σ is shown. The microtubule is either guided or dissociated. (Inset) Track geometry that imposes the constraint expressed by eq 2 on the microtubules. (b) Interaction between a microtubule and a kinesin motor. The green arrow indicates the direction of microtubule propulsion.

MT movement was simulated with Brownian dynamics under the constraint of fixed segment length.51 In this method, a single time step consisted of the following two steps. In the first step, the beads representing an MT were moved, without taking into account any constraint, using the following expression r′i (t + Δt ) = ri(t ) +

2. METHODS 2.1. Simulation Method. We simulated the 3D movements of MTs propelled by kinesin motors on microfabricated tracks (Figure 1a). In the case of the chemical edge, the boundary was defined to be on the x axis. MTs were subjected to the constraint z≥0

2DΔt N(0, 1)

(i = 1, ..., n)

(3)

where ζ is the viscous drag coefficient, Fi is the total external force applied to ri, D is the diffusion coefficient, N(0, 1) is a normalized random number with a mean of zero and a variance of 1, and n is the number of beads representing the MT. Δt was set at 10−6 s to ensure numerical stability. The viscous drag coefficient used was the average of the parallel and perpendicular drag coefficients48,52

(1)

In cases of kinesin-coated and nonfouling microfabricated walls, the microfabricated walls with height h and steepness σ were located parallel to the x axis, and their edges were placed on the x- axis. MTs were subjected to the following constraints (Figure 1a inset): ⎧ z ≥ 0, if y ≤ 0 ⎪ h ⎪ ⎪ z ≥ y tan(θ), if 0 ≤ y ≤ ⎨ tan(θ) ⎪ h ⎪ z ≥ h , if y ≥ ⎪ tan(θ) ⎩

Δt Fi + ς

ζ=

3πηd

( 2dr )

ln

(4)

where η is the viscosity of water (0.001 Pa s), d is the length of the MT segment (0.5 μm), and r is the MT radius (12.5 nm). The diffusion coefficient was calculated using D = kBT/ζ. The sources of Fi were assumed to originate from the kinesin motors (described below) and the bending of the MT. The restoring force of MT bending was calculated from the following bending potential48

(2)

U=

We assumed the MTs to be infinitely thin and inextensible semiflexible bead−rod polymers with a flexural rigidity of 22.0 pN μm2.50 The length of the MTs was set at 5 μm, in accordance with the experiment conducted by Clemmens et al.,30 and the MTs consisted of 10 rigid segments (Supporting Information).

1 EI 2 d3

n−1

∑ (ri+ 1 − 2ri + ri− 1)2 i=2

(5)

where EI is the flexural rigidity. In the second step, the unconstrained movements were corrected by taking into account the constraints due to the segment length and the guiding tracks. To keep the segment 12090

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component directed toward the minus end of the MT. Thus, the MT was propelled toward its minus end. For simplicity, by following the approach taken by Gibbons et al.,54 we neglected the spontaneous dissociation of the bound kinesin from the MT. That is, in the absence of an external force, the kinesin bound to the MT never dissociates unless it reaches the plus end of the MT. When a force is applied to the bound kinesin that exceeds the detachment force of 7 pN,55 the kinesin motor detaches from the MT. 2.2. Visualization. The simulation results were visualized with ParaView. 2.3. Calculation of Guiding Probabilities. We considered an MT arriving at a track boundary to be guided if it maintained contact with at least one kinesin motor on the bottom track and continued to travel on the bottom surface for more than 5 μm. If the MT lost contact with the bottom track surface by either dissociating or climbing up the microfabricated walls, then the MT was not considered to be guided. The probability of guiding was defined as the ratio of the number of guiding events to the total number of collision events as a function of the approach angle. Following the method of Clemmens et al.,30 standard errors were determined for the probabilities by the formula

length constant, the coordinates of the beads representing the MT {ri} were subjected to the following holonomic constraints: gsegment, k = (rk + 1 − rk)2 − d 2 = 0

(k = 1, ···, n − 1) (6)

To keep the MT movement within the regions expressed with eqs 1 or 2, the position of the beads representing the MT were subjected to the following holonomic constraints:

gtrack, i

⎧ zi = 0, if zi < 0 and y ≤ 0 i ⎪ ⎪ h ⎪ z − y tan(θ) = 0, if zi < y tan(θ) and 0 ≤ y ≤ i i =⎨ i tan(θ) ⎪ h ⎪ ⎪ zi − h = 0, if zi < h and yi ≥ tan(θ) ⎩

(7)

The correction was carried out with the following expression ri(t + Δt ) = r′i(t + Δt ) + Δri(t + Δt )

(8)

where Δri′(t + Δt) is the correction term: Δri(t + Δt ) =

Δt ζ

n−1

∑ λsegment,k k=1

∂gsegment, k ∂ri

+ λtrack, i

∂g track, i ∂ri (9)

λsegment,k and λtrack,i are Lagrangian multipliers, which were determined in order for the coordinates at t + Δt to satisfy the constraints expressed in eqs 6 and 7, respectively. To achieve this, we cyclically repeated the calculations for each of the constraints until the constraints were satisfied with a tolerance of 10−6 μm. Kinesin motors were randomly distributed over the allowed surfaces by specifying the positions of the kinesin tails. Unless stated otherwise, the surface density of the kinesins was 30 μm−2, in accordance with the experiment conducted by Moorjani et al.31 If an MT segment came close to a kinesin motor tail within a defined capture radius (w = 20 nm),53 then the kinesin motor was assumed to bind to the MT segment, and the position of the motor head was specified on the MT segment. The position of the motor head was chosen as the closest point to the kinesin tail. Hence, the pulling force from the kinesin was perpendicular to the MT segment at the moment of binding. The bound kinesin acted as a linear spring between the motor head and tail with a spring constant of 100 pN μm−1 and an equilibrium length of zero and exerted a pulling force on the MT segment. The pulling force was divided into two forces, which acted on the two beads located at either end of the MT segment where the kinesin motor was bound, under the condition that the total force and torque on the segment remained the same. The head of the bound kinesin motor moved toward the plus end of the MT (Figure 1b) with a force-dependent velocity expressed as ⎛ F ⎞ v(F ) = v0⎜1 − ⎟ Fstall ⎠ ⎝

SE =

p(1 − p) N

(11)

where p is the probability of guiding and N is the total number of MTs arriving at the boundary with an approach angle in that bin.

3. RESULTS AND DISCUSSION 3.1. Guiding by a Chemical Edge. The guiding of MTs by a chemical edge was simulated (Figure 2). On the kinesin track, MTs were translated with a speed of 0.792 ± 0.008 μm s−1, which is equal to the maximum translation speed of the kinesin motors (v0). During translation, the MTs showed fluctuations in their sliding directions. The variance of the

(10)

where v0 is the translational velocity in the absence of an applied force, F∥ is the component of the pulling force along the MT, and Fstall is the stall force of the kinesin motors. v0 was set at 0.8 μm s−1, and Fstall was set at 5 pN. Through this translation toward the plus end of the MT, the bound kinesin motor was elongated, building up a pulling force with a

Figure 2. Series of snapshots of microtubule movement guided by a chemical edge over time. The microtubule is represented by the orange line. The kinesin motors are represented by the yellow dots. The length of the microtubule is 5 μm. Scale bar: 3 μm. 12091

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directional change increased linearly with the time interval (Supporting Information Figure 1). The rate of increase was (7.0 ± 0.6) × 10−4 s−1, corresponding to a path persistence length of 1.12 ± 0.08 mm. This was shorter than the expected value of 5 mm50 because according to Duke et al.53 the path persistence length should be equal to the persistence length of MT itself. Presumably, the short path persistence length arose from pulling the kinesins bound to the MTs. When an MT arrives at the boundary, the leading tip protrudes from the kinesin track. The protruding part of the MT displayed thermal fluctuations. During these fluctuations, if the MT encountered a kinesin motor on the track, then it was guided (Supporting Information Movie 1). Otherwise, the MT continued to move off of the track, finally losing contact with the kinesin motors and dissociating from the track (Supporting Information Movie 2). Figure 3 shows the probability of guiding the MTs at various approach angles by the chemical edge. The probability is an Figure 4. Zipping mechanism of microtubule guiding by a chemical edge. The orange line represents the microtubule. The white dots represent kinesin motors. The green dots represent kinesin motors bound to the microtubule. The blue region indicates the track surface. The length of the microtubule is 5 μm. Scale bar: 2 μm.

mechanisms, we first investigated the fraction of MTs guided by swiveling. We considered swiveling to occur if an MT was held by a single kinesin.56 Among all of the MT trajectories guided by the chemical edge, for motor densities of 10, 30, and 50 μm−2, swiveling was determined to be 40, 12, and 18%, respectively. However, the movements of MTs appeared to be similar to those through the zipping (Supporting Information Movie 4). To investigate the reattachment of dissociated filaments (Supporting Information Movie 5), by matching the capture rate of common microscope observations, we generated MT trajectories with capture rates of 10 and 1 Hz and calculated the probabilities of guiding for the trajectories with low capture rates (hence, we systematically ignored short dissociations). Changing the capture rate did not significantly alter the guiding probabilities (Supporting Information Figure 3). Thus, the reattachment of dissociated filaments does not contribute significantly to the discrepancy between the theoretical prediction and the experimental results. Good agreement in the guiding probability with the experimental results was also achieved with the 2D model by Rupp and Nedelec.49 This is plausible in light of the following observation: In our simulation, MTs remained 3.6 ± 3.6 nm above the substrate; the part of the MT extending over the kinesin-free area was 23 ± 23 nm above the substrate (Supporting Information Figure 4). Because MTs spent most of the time within the capture radius of 20 nm, the 2D model well reproduces the guiding by the chemical edge. 3.2. Guiding by a Microfabricated Wall Coated with Kinesin. The guiding of MTs by a microfabricated wall completely coated with kinesin was simulated. Figure 5 shows a time series of an MT being guided by a microfabricated wall completely coated with kinesin. MTs bent when they reached the microfabricated walls. If the tips bound to kinesin motors were located either on the wall or on the bottom surfaces, then the MTs continued to move on the surface where the bound kinesin motor was located (Supporting Information Movies 6 and 7).

Figure 3. Probability of guiding for various microtubule approach angles at a chemical boundary (mean ± standard error). The probability of guiding was obtained from 427 simulated microtubule paths. The red dashed curve represents the experimental fit reported by Clemmens et al.;30 y = 1.2 exp(−0.064x).

exponentially decreasing function of the approach angle, which is in good agreement with the experimentally obtained probability function reported by Clemmens et al.30 The probability of guiding was not significantly altered by changing the motor density from 10 to 50 μm−2 (Supporting Information Figure 2a). A closer look reveals that MTs are guided through the “zipping” of kinesin motors binding to the MTs (Figure 4 and Supporting Information Movie 3). The part of the MT extending over the kinesin-free area exhibits thermal fluctuations, during which this part of the MT closely approaches the track boundary and binds to kinesin motors near the pivot point. This binding biases the thermal fluctuations of the MT closer to the track boundary, rendering other kinesin motors located along the track boundary more likely to bind to the MT. Thus, a series of binding events propagates from the pivot point to the leading tip of the MT. This guiding mechanism is in contrast to the assumption of the analytical models of Clemmens et al.30 and Sundberg et al.,35 who assumed that guiding occurred when the MT leading tip reached a new kinesin. This may explain the discrepancy between the theoretical guiding probability obtained by Sundberg et al. and the experimentally obtained guiding probability. Sundberg et al. pointed out two possibilities for this discrepancy: the swiveling of filaments and the reattachment of dissociated filaments. To further explore these two 12092

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microfabricated wall and thereafter moved on the bottom track (Supporting Information Movie 8). Figure 7 shows the probability of MT guiding by a perfect nonfouling wall (i.e., with no kinesin bound to the wall). The

Figure 7. Probabilities of guiding for various microtubule approach angles at nonfouling microfabricated walls. The steepness of the walls was 90°. The solid squares show the probability of guiding by a perfect nonfouling wall, and the open squares show the probability of guiding by a nonfouling wall with the unintended adhesion of kinesin with a density of 3 μm−2 (mean ± standard error). For the perfect nonfouling wall, the probability of guiding was obtained from 213 simulated MT paths; for the nonfouling wall with an unintended adhesion of kinesin, the probability was obtained from 214 simulated MT paths. The red dashed line is the experimental fit reported by Clemmens et al.;30 y = 0.87. For better visibility, the plots at 5° and 15° were shifted horizontally.

Figure 5. Time series of snapshots of a microtubule guided by a microfabricated wall completely coated with kinesin. The orange line represents the microtubule. The yellow dots represent the kinesin motors. The length of the MT was 5 μm.

Figure 6 shows the probability of MT guiding by a microfabricated wall with a steepness of 85°. (The steepness

probability was 1.0 regardless of the approach angle, showing a slight discrepancy with the values reported by Clemmens et al.30 The probability was independent of the motor density on the track (Supporting Information Figure 2c). In investigating the reason for the discrepancy, we found that when kinesin motors adhered to the microfabricated wall with a density of 3 μm−2, the guiding probability decreased to 0.92, which is in agreement with the experimental results (Figure 7). This result indicates that a small number of kinesin motors adhered to the poly(ethylene oxide)-coated surface, which is consistent with results from Katira et al.57 Van den Heuvel et al. experimentally measured the probabilities of MT guiding by gold tracks with poly(ethylene glycol)-coated SiO2 walls.58 Although the guiding method was essentially the same as that of Clemmens et al.,30 they obtained different guiding probabilities: the probability of guiding was 1.0 below a critical approach angle and then decreased almost linearly with increasing approach angle. A similar guiding probability was also reported by Lin et al.36 To understand the reason for this discrepancy, we changed the steepness of the guiding walls. By decreasing the steepness, we obtained guiding probabilities similar to those obtained by van den Heuvel et al. and Lin et al. (Figure 8 and Supporting Information Figure 6 for the steepness range from 70 to 90°). This result indicates that differences in the wall steepness lead to differences in the guiding probability. Lin et al.36 rationalized their experimentally obtained guiding probability by developing a model based on statistical mechanics. The guiding probability reported in their study was different from those obtained from our simulation, even for the same kinesin density. This discrepancy may be resolved by the following two observations: First, MT tips often slide down the side walls and bind to kinesin motors on the bottom track (Supporting Information Figure 7). Such slipping was not considered in their analytical model. Second, the length of the

Figure 6. Probability of guiding for various microtubule approach angles at a microfabricated wall coated with kinesin (mean ± standard error). The probability of guiding was obtained from 344 simulated MT paths. The steepness of the wall was 85°. The red dashed curve is the experimental fit reported by Clemmens et al.;30 y = 0.64 − 0.0067x.

of the microfabricated wall created by Clemmens et al. was reported to be 70−80°.) The probability decreases as a function of the approach angle, and the values are in agreement with those obtained by Clemmens et al.30 (For different motor densities, see Supporting Information Figure 2b). We also obtained the probabilities of guiding for walls of different steepnesses, covering the realistic steepness range for microfabricated walls (Supporting Information Figure 5). With decreasing steepness, the probability of guiding decreased sharply, indicating that precise control of the steepness is required in fabricating the walls. 3.3. Guiding by Nonfouling Microfabricated Walls. The guiding of MTs by a nonfouling microfabricated wall was simulated. Gliding MTs were bent upon contacting the 12093

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This simulation could be easily extended to actin/myosinbased molecular shuttles as well as shuttles with microtubuleactin filament hybrid filaments59 and fascin-cross-linked actin bundles.60 Combining this simulation with our previous simulations37,38 would be a powerful strategy for designing microdevices powered by motor proteins. In the previous simulations, guiding probabilities had to be measured experimentally and implemented into the simulation. The simulation described here enables the prediction of guiding probabilities, which makes the design methodology less dependent on experiments. Hence, this can be viewed as a step toward a full computeraided design for microdevices powered by motor proteins.

Figure 8. Effect of wall steepness on the probability of guiding by perfect nonfouling walls. The solid squares correspond to a steepness of 75°, and the open squares correspond to a steepness of 85° (mean ± standard error). For a wall with a steepness of 75°, the probability of guiding was obtained from 236 simulated MT paths; for a wall steepness of 85°, the probability was obtained from 225 simulated MT paths. For better visibility, the plots at 5, 15, 35, and 45° were shifted horizontally.

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ASSOCIATED CONTENT

S Supporting Information *

Variance of directional change of gliding microtubules. Probabilities of guiding with various surface densities of kinesin motors. Effect of reattachment of dissociated microtubules on probabilities of guiding at chemical edges. Height distributions of microtubules from substrates. Effect of the wall steepness on probabilities of guiding at microfabricated walls coated with kinesin. Effect of the wall steepness on probabilities of guiding at nonfouling microfabricated walls. Slipping of the leading tip of a microtubule on a nonfouling guiding wall. On the number of the segments representing microtubules. Movies of microtubule movements. This material is available free of charge via the Internet at http://pubs.acs.org.

bent parts of the MTs was longer than that assumed by Lin et al. They assumed that the length of the bent parts of the MTs was equal to the distance between the MT leading tip and the foremost kinesin. Our simulation revealed that when an MT reached the wall and was bent, some kinesin motors detached from the MT, relaxing the MT bending and allowing new bindings to occur (Figure 9 and Supporting Information Movie 9). This indicates that the analytical model of Lin et al. may overestimate the bending energy of the MTs.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Author Contributions

The manuscript was written through the contributions of both authors. Both authors have given approval to the final version of the manuscript. Notes

The authors declare no competing financial interest.

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Figure 9. Superimposed snapshots of a microtubule approaching a perfect nonfouling wall. The orange lines show the conformations of the microtubule with an interval of 0.2 s. The white dots represent the kinesin motors. The green dots represent the kinesin motors bound to the microtubules. The green arrows indicate the directions of the microtubule movement. Scale bar: 1 μm.

ACKNOWLEDGMENTS This work was supported by JSPS Grant-in-Aid for Young Scientists (B) (grant number 24760203). We gratefully acknowledge Amy Lam and Henry Hess of Columbia University for valuable discussions and critical comments on the manuscript.

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4. CONCLUSIONS Using the simulation developed in this study, we reproduced the 3D motion of MTs at various types of boundaries and investigated the mechanisms by which MTs were guided. The simulation had two key advantages: (1) enabling analysis with high spatiotemporal resolution and (2) allowing precisely controlled conditions. The former allowed us to discover a zipping mechanism of kinesin binding to the extended tips of MTs at a chemical edge; the latter allowed us to investigate the effects of unintended adhesion of kinesin on the nonfouling wall, as well as the effect of wall steepness on MT guiding. These results would have been difficult to obtain using analytical models or through experiments. Hence, this demonstrates the usefulness of this simulation for investigating the mechanisms of guiding, interpreting experimental results, and testing the assumptions upon which the analytical models are based.

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